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Chapter 9 Theory of Differential and Integral Calculus• Section 1 Differentiability of a Function
Definition. Let y = f(x) be a function. The derivative of f is the function whose value at x is the limit
provided this limit exists.
If this limit exists for each x in an open interval I, then we say that f is differentiable on I.
E.g.1.2 (a) Show that f(x)=|tanx| is not differentiable at x=n*Pi
Proof of e.g.1.2
.at x abledifferentinot is f(x)
-1 xcos
1lim
h
sinhlim)(f Similarly,
1 11 xcos
1lim
h
sinhlim
h
|tanh|lim
h
|)tan(||)htan(|lim
h
)(f)h(flim)(f )a(
0h0h-
0h0h0h
0h
0h
0.at x abledifferentinot is xg(x) that Show )b( 3
2
h
1lim
h
)0(g)h0(glim ,fact In
0at x abledifferentinot is g(x)
h
1lim
h
0)h0(lim
h
)0(g)h0(glim
3
10h0h
3
10h
3
2
0h0h
Discussion: Ex.9.1, Q.1, 6Discussion: Ex.9.1, Q.1, 6
Solution to Ex.9.1Q.1
-1b i.e.
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Discussion Ex.9.1, Q.9Discussion Ex.9.1, Q.9
Section 2 Mean Value Theorem
Rolle's Theorem. Let f be a function which is differentiable on the closed interval [a, b]. If f(a) = f(b) then there exists a point c in (a, b) such that f '(c) = 0.
Example of Rolle’s Theorem
Mean Value Theorem
Mean Value Theorem. Let f be a function which is differentiable on the closed interval [a, b]. Then there exists a point c in (a, b) such that
ab
)a(f)b(f)c('f
Proof of Mean Value Theorem
(Q.E.D.) a-b
f(a)-f(b)(c) f.e.i
0a-b
f(a)-f(b)-(c) f i.e.
0.(c)' such that b) ,a(c
Theorem, sRolle' by the and (b)(a)ab
)b(af)a(bfb
a-b
f(a)-f(b)-f(b)(b) and
ab
)b(af)a(bfa
a-b
f(a)-f(b)-f(a)(a) then
,xa-b
f(a)-f(b)-f(x)(x)function theConsider
Example 1 of Mean Value Theorem
Example 2 of Mean Value Theorem
Corollaries
(1) Let f be a differentiable function whose derivative is positive on the closed interval [a, b]. Then f is increasing on [a, b].
(2) Let f be a differentiable function whose derivative is negative on the closed interval [a, b]. Then f is decreasing on [a, b].
Proof of Corollary(1)
increasing is f(x) hence and )x(f)x(f
0)c('fxx
)f(x-)f(xsuch that
b)(a,c exists thereb,xxaany For
12
12
12
21
decreasing is f(x) hence and )x(f)x(f
0)c('fxx
)f(x-)f(xsuch that
b)(a,c exists thereb,xxaany For
12
12
12
21
Proof of corollary(2)
First Derivative Test. Suppose that c is a critical point of the function f and suppose that there is an interval (a, b) containing c. (1) If f '(x) > 0 for all x in (a, c) and f '(x) < 0 for all x in (c, b), then c is a local maximum of f. (2) If f '(x) < 0 for all x in (a, c) and f '(x) > 0 for all x in (c, b), then c is a local minimum of f.
Corollary 3If f ’(x)=0 for all x in an interval I, then f(x) is a constant function in I.
I.in function constant a is f
)x(f)f(x
0)c(fxx
)x(f)f(x
such that I)x,(xc Theorem, ValueMean by
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12
21
21
Corollary 4If f and g are differentiable functions on I and f ’(x)=g’(x) for all x in I, then f(x)=g(x) +c for some constant c.
I.on c g(x)f(x) thus
and Rc , ch(x) 2,corollary By
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Section 3 Convex Functions
Definition 3.1
• A real-valued function f(x) defined on an interval I is said to be convex on I iff
• for any two points x1, x2 in I and any two positive numbers p and q with p+q=1,
• f(px1+qx2) pf(x1) + qf(x2).
I.on convex is f then 0,(x) f If
3.1 Theorem
Example
ab2
ba.e.i
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.Ron convex is f ,0x
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Theorem 3.2
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Proof of Theorem 3.2
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§4 Definite Integral is the Limit of a Riemann Sum
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1
n
2
n
i
n
n
n
y=f(x)Find the sum of the areas of the rectangles in terms of n and f.
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f
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f
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y=f(x)
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:Example
Group DiscussionExpress each of the following integrals as a limit of sum of areas:
1
0
x
1
0
1
0
2
1
0
dxe .4
sinxdx .3
dxx .2
xdx .1
2222n n
n...
n
3
n
2
n
1lim
3
2
3
2
3
2
3
2
n n
n...
n
3
n
2
n
1lim
n
1in n
isin
n
1lim
n
1i
n
i
ne
n
1lim
Group DiscussionExpress each of the limits as a definite integral :
1
0
x-
1
0
1
0
1
0
3
dxe
cosxdx
dxx1
1
dxx
4
3
4
3
4
3
4
3
n n
n...
n
3
n
2
n
1lim .1
nn
1...
3n
1
2n
1
1n
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n
1in n
icos
n
1lim .3
n
1i
n
i
ne
n
1lim .4
4
1
2ln
1sin
1e1
Example 4.1(a)
4
π=]x[tan=
x+1
dx=
nn
+1
1+...+
n3
+1
1+
n2
+1
1+
n1
+1
1
n
1lim=
n+n
1+...+
3+n
1+
2+n
1+
1+n
1n lim
:Solution
10
110 2
22
22
22
22∞→n
22222222∞→n
∫
.nn
1...
3n
1
2n
1
1n
1nlim Evaluate )a(
22222222n
Read example 4.1(b)
Classwork Ex.9.4 Q.3
Read example 4.1(b)
Classwork Ex.9.4 Q.3
Area bounded by the curve, x-axis, x=a and x=b
a bn
aba
n
)ab(2a
n
)ab(3a
n
)ab)(1n(a
n
abaf
n
)ab(2af
n
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)b(f
n
abrectangles of width
n
1in
b
a
n
abiaf
n
a-blim
dxf(x)
b and abetween curve under the area the
Homework Ex.9.4Homework Ex.9.4
Example 4.1(a)
4
π=]x[tan=
x+1
dx=
nn
+1
1+...+
n3
+1
1+
n2
+1
1+
n1
+1
1
n
1lim=
n+n
1+...+
3+n
1+
2+n
1+
1+n
1n lim
:Solution
.n+n
1+...+
3+n
1+
2+n
1+
1+n
1nlim Evaluate )a(
10
110 2
22
22
22
22∞→n
22222222∞→n
22222222∞→n
∫
n
1kn
b
a n
)ab(kaf
n
ablimdx)x(f
Section 5 Properties of Definite Integrals
.0dxf(x) then b], ,a[x 0,f(x)
and b] [a,on integrable is f(x) If 5.1 Theoremb
a
0n
)ab(kaf
n
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n
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When does the equality fail?When does the equality fail?
Theorem 5.4Theorem 5.4
Discussion:Ex.9.5, Q.1,2Discussion:Ex.9.5, Q.1,2
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
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Corollary 5.5Corollary 5.5
Discussion: Ex.9.5, Q.3Discussion: Ex.9.5, Q.3
Corollary 5.3
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a
b
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b
a
b
a
b
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Homework:Ex.9.5,3-7Homework:Ex.9.5,3-7
Example 5.2
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1..g(0) and 0f(0) C.
R;for x -f(x)(x)g B.
R;for x g(x)(x)f A.
:conditions following the
satisfying Ron defined functions abledifferenti be g and f Let
Rfor xcosx g(x) andsinx f(x) that show otherwise, or
,]xcos)x(g[sinx]-[f(x)h(x) atingdifferentiBy 22
R.for xcosx g(x) andsinx f(x)
0cosx-g(x) and 0xsin)x(f
0]xcos)x(g[sinx]-[f(x)
0h(x)
0
1]-[10
]0cos)0(g[sin0]-[f(0)h(0)c
get we0, xPutting
c.constant somefor c)x(h
0
)xsin)x(f)(xcos)x(g(2
)xcos)x(g)(xsin)x(f(2
)xsin)x(g)(xcos)x(g(2
)xcos)x(f)(xsin)x(f(2)x(h
:Solution
22
2
22
Example 5
..dx)x(ff(x)dx ab
thenf(a),b0 if that (a) from Deduce (b)
a]. [0,u allfor 0F(x) that Prove (a)
).u(ufdx)x(f dx)x(f F(x)Let
0.f(0) and 0a where
a], [0,on function increasingstrictly and abledifferenti a is f
a
0
b
0
1-
f(u)
0
1-u
0
a]. [0,u allfor 0F(x) that Prove (a)
).u(ufdx)x(f dx)x(f F(x)Let f(u)
0
1-u
0
a
f(a)
u
f(u)
b
u
0dx)x(f
f(u)
0
1- dx)x(f
0
)u(f)u(fu)u(fuf(u)
)u(f)u(fu)u(f))u(f(f)u(f)u(F
:oofPr1
a] [0,u allfor 0F(x)
0
)0(f0dx)x(f dx)x(f F(0)
get we0, xPuttingf(0)
0
1-0
0
c.constant somefor cF(x)
a
0
b
0
1- .dx)x(ff(x)dx ab
thenf(a),b0 if that (a) from Deduce (b)
f(u)
0
1-u
0).u(ufdx)x(f dx)x(f F(x)Let
0
)u(f)u(fu)u(fuf(u)
)u(f)u(fu)u(f))u(f(f)u(f)u(F
:oofPr1
a] [0,u allfor 0F(x)
0
)0(f0dx)x(f dx)x(f F(0)
get we0, xPuttingf(0)
0
1-0
0
c.constant somefor cF(x)
a
f(a)
u
f(u)
b
a
0dx)x(f
b
0
1- dx)x(f
0
0u
∫∫∫∫∫
∫∫
)f(u0 00
1-u0
au
)f(u0 000
1-u0
00
f(u)0
-1u0
000
00
.b)ua(+bu>dx)x(f +dx)x(f +dx)x(f
.bu=)u(fu=dx)x(f +dx)x(f
then,b=)f(usuch that u=u Let
).u(uf=dx)x(f +dx)x(f (a), By
ab=